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### Homotopy limit of a cosimplicial category

Consider the usual model structure on Cat (category of small categories).
Which are the fibrations of the injective model structure on the category of cosimplicial categories $Fun( \Delta ,Cat )$?
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### Joins of, and limits in, $(\infty,1)$-categories via profunctors

I'm trying to interpret the join of $(\infty,1)$-category in a more conceptual way. Let me try to explain what I have in mind.
In the classical setting it is almost a triviality to express the join ...

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### About elegant Reedy categories

I discovered today the notion of elegant Reedy category introduced in the paper Reedy categories and the $\Theta$-construction of Julia E. Bergner and Charles Rezk. An interesting property of such ...

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### Simplicial localisation and infinity categories

If $(\mathcal{C},W)$ is a category with weak equivalences then we may naturally form its Dwyer-Kan simplicial localisation $L(\mathcal{C}, W)$. This is a simplicial category which naturally gives a ...

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### K theory of a simplicial monoidal category, Cofinality theorem

Let $X=(d\mapsto X_d)$ be a simplicial symmetric monoidal category. We define the $K$-theory space of $X$ to be $K(X)=|d\mapsto K(X_d)|$, the geometric realisation of the simplicial space $d\mapsto ...